Optimal stability for inverse elliptic boundary value problems with unknown boundaries

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

G IOVANNI A LESSANDRINI

E LENA B ERETTA

E DI R OSSET

S ERGIO V ESSELLA

Optimal stability for inverse elliptic boundary value problems with unknown boundaries

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 29, n

^o

4 (2000), p. 755-806

<http://www.numdam.org/item?id=ASNSP_2000_4_29_4_755_0>

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Optimal Stability ^{for Inverse} Elliptic Boundary

Value Problems with Unknown Boundaries

GIOVANNI ALESSANDRINI - ELENA BERETTA - EDI ROSSET - SERGIO VESSELLA

Abstract. In this paper ^we study ^{a class} of inverse problems associated to elliptic boundary ^value problems. ^More

precisely,

those inverse problems in which the role of the unknown is

played

by ^an inaccessible part ^{of the} boundary ^{and the role}

of the data is

played

by overdetermined boundary ^{data for the} elliptic equation assigned ^{on the}

remaining,

accessible, part ^{of the} boundary. ^{We treat the case}

of

arbitrary

space dimension n &#x3E; 2. Such problems ^{arise in}

applied

^contexts

of nondestructive testing of materials for either electric or thermal conductors, and are known to be ill-posed. ^In this paper ^we obtain essentially ^best possible

stability

estimates. Here, in the context of

ill-posed

problems, stability ^{means the}

continuous dependence of the unknown upon the data when additional ^a

priori

information on the unknown boundary (such ^{as its} regularity) is available.

Mathematics Subject Classification (2000): ^35R30 (primary), 35R25, 35R35, 35B60, 31B20 (secondary).

1. - Introduction

In this paper ^we shall deal with two inverse

boundary

^value

problems.

Suppose Q

^{is a} bounded domain in W with

sufficiently

smooth bound- ary

a S2,

^a

part

^of

which,

say ^I

(perhaps

^some interior connected

component

of a S2 or some inaccessible

portion

of the exterior

component

^of

a03A9),

^{is not}

known. This could be the case of an

electrically conducting specimen,

^{which is}

possibly

defective due ^to the presence of interior cavities ^or of corroded

parts,

which are not accessible to direct

inspection.

^See for instance

[K-S-V].

^{The aim}

is to detect the presence of such defects

by

nondestructive methods

collecting

current and

voltage

measurements on the accessible

part

^{A of the}

boundary

If we assume that the inaccessible

part

^{I of a 03A9 is}

electrically ^insulated, then, given

^a nontrivial

function 1/1

^on

^A, having

^zero average

(which represents

Work

supported

ⁱⁿ part by ^MURST.

Pervenuto alla Redazione il 21 settembre 1999 e in forma definitiva il 24

giugno

^2000.

Vol. XXIX (2000), pp. 755-806

the

assigned

^current

density

^on the accessible

part

^{A of}

8Q),

^{we have that the}

voltage potential

^u inside Q satisfies the

following

^Neumann

type boundary

value

problem

Here,

^{v is} the exterior unit normal to aS2 and cr ⁼ denotes the known

symmetric conductivity

tensor ant it is assumed to

satisfy

^a

hypothesis

of uniform

ellipticity.

^{Let us} remark that the solution to

( 1.1 a )-( 1.1 c)

^is

unique

up ^{to an} undetermined additive constant. In order to

specify

^a

single ^solution,

we shall _assume, from now on, the

following

normalization condition

Suppose,

^now, that E is an open subset of which is contained in

A,

^and

on which the

voltage potential

^can be measured.

Then,

the inverse

problem

consists of

determining

^I

provided

is known. This is the first

object

^{of our}

study

^{and we} shall refer to it as the Inverse Neumann Problem

(Neumann

^case,

for

short).

An allied

problem

^{is the one} associated to the direct Dirichlet

problem

Here,

^as

above, I,

^{A are the}

inaccessible, respectively, accessible,

_parts ^of and ar is the

conductivity

^tensor

satisfying

^{the same}

hypotheses.

^{Our second}

object

^of

study

is the inverse

problem consisting

in the determination of I from the

knowledge

^{of orVM’ where E C A is as} above. We shall refer to it as

the Inverse Dirichlet Problem

(Dirichlet

^{case, for}

short).

^We believe that also this

problem

may be of interest for concrete

applications

^of

nondestructing testing,

for instance in thermal

imaging.

^{In this case, the} inaccessible

boundary

^I

could

represent

^a

priviledged

isothermal

surface,

^{such as a} solidification front.

Of _course, it should be

kept

^{in mind}

^that, dealing

with thermal processes, the

evolutionary

model based on

parabolic,

rather than

elliptic, equations

^{is in}

general

^more

appropriate,

for related issues see, for

instance, [B-K-W], [Bi], [V 1 ] . However,

^{we trust} that also a

preliminary study

^{of a}

stationary

^model

may be instructive.

Such two

problems,

the Neumann and Dirichlet cases, ^are known to be

ill-posed.

Indeed there are

examples

^{that show}

^that,

^{under a}

priori assumptions

on the unknown

boundary

^I

regarding

^its

regularity (up

^to any finite order of

differentiability),

the continuous

dependence (stability)

^{of I} from the measured data in the Neumann

case, orVM’

_VIE in the Dirichlet

case) is,

^at

best,

^of

logarithmic type.

^See

[A12]

^{and also}

[Al-R].

The main purpose of this paper is ^to prove

stability

estimates of

loga-

rithmic

type (hence,

^best

possible)

for both the Neumann and Dirichlet cases,

(Theorems 2.1, 2.2),

when the space

dimension n &#x3E;

^{2 is}

arbitrary.

^{We recall}

that,

^{for the case n =}

2,

^results

comparable

^{to ours} have been found in

[Be-V]

when o~ is

homogeneous

^{and in}

[R], [Al-R]

^{when o~ can be}

inhomogeneous

and also discontinuous. Other related results for the case of dimension two can

be found in

[Bu-C-Yl], [Bu-C-Y2], [Bu-C-Y3], [Bu-C-Y4], [An-B-J].

^{Let us}

also recall

that, typically,

the above mentioned results are based on

arguments related,

ⁱⁿ various ways, ^to

complex analytic ^methods,

^{which do not} carry ^over the

higher

dimensional case.

In the

sequel

^{of this}

Introduction,

^we shall illustrate the new tools we

found _necessary to

develop

^and

exploit

^{when n} &#x3E; 2. But

first,

^{let us comment}

briefly

^on the connection with another inverse

problem

which has become

quite popular

in the last ten years,

namely

the inverse

problem

of cracks. On one

hand there are

similarities,

in fact a crack can be viewed as a

collapsed cavity,

that is a

portion

of surface inside the

conductor,

such that a

homogeneous

Neumann condition like

(I,lc)

^{holds on the two} sides of the surface. On the other hand there are

differences,

^{for the}

uniqueness

in the crack

problem

^at

least two

appropriate

^distinct measurements are necessary

[F-V],

whereas for

our

problems,

either the Neumann or the Dirichlet _case, any

single

^nontrivial

measurement suffices for

uniqueness,

^see for instance

[Be-V]

^{for a discussion}

of the

uniqueness

^{issue. Let us} also recall that for the crack

problem

ⁱⁿ

dimensions

bigger

than two, various basic

problems regarding uniqueness

^are

still unanswered.

See,

for the available results and for references

[Al-DiB ] .

^{It is}

therefore clear that a

study

^{of the}

stability

for the crack

problem

in dimensions

higher

^{than two shall}

require

^{new ideas.}

Nonetheless,

the authors believe that the

techniques developed

^here

might

be useful also in the treatment of the crack

problem.

The methods we use in this paper ^are based

essentially

^{on a}

single unifying

theme:

Quantitative

^Estimates

of Unique Continuation,

^{and we shall}

exploit

^it

under various different

facets, namely

^the

following

^ones.

(a) Stability

^Estimates

of

Continuation

from Cauchy

^{Data. Since we are}

given

the

Cauchy

^{data on E for a solution u to}

(l.la),

^we shall need to evaluate how much a

possible

^{error on such}

Cauchy

^{data can} affect the interior values of u. Such

stability

estimates for

Cauchy problems

associated to

elliptic equations

^{have been a central}

topic

^of

ill-posed problems

^{since the}

beginning

of their modem

theory, [H], [Pul], [Pu2]. Here,

^{since one of}

our

underlying

aims will be to treat our

problems

^under

possibly

^minimal

regularity assumptions,

^{we shall assume the}

conductivity

^{cr to be}

Lipschitz

continuous

(this

is indeed the minimal

regularity ensuring

^the

uniqueness

for the

Cauchy problem, [PI], [M]).

^Our

_present stability

^estimates

(Propo-

sitions

3.1, 3.2, 4.1, 4.2)

will elaborate on

inequalities

^{due to}

Trytten [T]

who

developed

^a method first introduced

by Payne [Pal], [Pa2].

^{The ad-}

ditional

difficulty

encountered here will be that we shall need to compare solutions _{u 1, u 2} which are defined on

possibly

different domains

Q2

whose boundaries are known to agree ^on the accessible

part A only.

^Let

us recall that ^a similar

approach,

but restricted to the

topologically simpler

two-dimensional

setting,

^has

already

been used in

[All], [Be-V].

^{We shall}

obtain

that,

^{if the error on the} measurement on the

Cauchy

^{data is}

^small,

then for the Neumann case,

also IVUII I

^is

small,

^{in an}

L2

_{average sense,}

on

S21 B Q2,

^the

part

^of

S21

which exceeds

Q2.

^{And the same holds for}

on

SZ2 BQI (Propositions ^3.1, 3.2).

In the Dirichlet case instead we

shall prove that ^{u 1} itself is small in

S21 B Q2,

^{and the same holds for} u2 on

S22 (Propositions ^{4.1, 4.2).}

(b)

^Estimates

of

Continuation

from

the Interior. We shall also need interior average lower bounds ^{on u} and on its

gradient (Propositions ^3.3, 4.3),

^on

small balls contained inside S2. Bounds of this

type

have been introduced in

[Al-Ros-S,

^Lemma

2.2]

^{in the context of a} different inverse

boundary

value

problem.

^The tools here involve another form of

quantitative unique continuation, namely

^the

following.

(c)

^Three

Spheres Inequalities.

^{Also this one is a} rather classical theme in connection with

unique

continuation. Aside from the classical Hadamard’s three circles

theorem,

^{in the context of}

elliptic equations

^we recall Lan- dis

[La]

^and

Agmon [Ag].

^{Under our}

assumptions

^of

Lipschitz continuity

on a, ^our estimates

(see (5.47) below)

shall refer to differential

inequalities

on

integral

^norms

originally

^{due to} Garofalo and Lin

[G-L],

^later

developed by

Brummelhuis

[Br]

and Kukavica

[Ku].

(d) Doubling Inequalities

^{in the Interior.} This rather recent tool has been intro- duced

by

Garofalo and Lin in the above mentioned paper

[G-L].

^It

provides

an efficient method of

estimating

^the local average

vanishing

^{rate of a so-}

lution to

(l.la).

^{Let us} recall that it also

provides

^a remarkable

bridge

to the

powerful theory

^of

Muckenhoupt weights ^[C-F]

and that this last connection has been

crucially

^{used in}

[Al-Ros-S]

and also in

[V2].

The

last, fundamental,

appearance of

quantitative

estimates of

unique

^con-

tinuation is the

following.

(e) Doubling Inequalities

^{at the}

Boundary.

^{For our} purposes it will be crucial to evaluate the

vanishing

^{rate of Vu}

(in

the Neumann

case)

^{or of u}

(in

the Dirichlet

case)

^near the inaccessible

boundary

^{I. In}

particular,

^{the fact}

that such a rate is not worse than

polynomial (Propositions ^3.5, 4.5)

^{is an}

essential

ingredient

ⁱⁿ

proving

^{that the}

stability

^{for our inverse}

problems

^are

not worse than

logarithmic (see

^the

proof

of Theorem

2.1).

Such evalua- tions on

vanishing

^{rates near}

^I,

^{where an}

homogeneous boundary

^condition

applies (either (I,lc)

^or

( 1.2. c) ),

^{can be obtained}

by

^{the so} called Dou-

bling Inequalities

^{at the}

Boundary.

^The

study

^{of such}

inequalities

^{has been}

initiated

by Adolfsson,

Escauriaza and

Kenig [A-E-K]

^{and later}

developed by

Kukavica and

Nystrom [Ku-N]

and Adolfsson and Escauriaza

[A-E].

^In

particular,

ⁱⁿ

^[A-E]

^such

inequalities

^are proven, for the Neumann

problem,

when the

boundary

^{is 1}

smooth, and,

for the Dirichlet

problem,

^when

the

boundary

^is

^smooth,

where the modulus of

continuity w

^{is of}

Dini

type.

^{We shall use}

essentially

^such

^results,

^{with the}

only

^difference

that, mainly

^for

simplicity

^of

exposition,

^{we shall assume, in the Dirichlet}

case, that

w (t)

⁼

t",

⁰ a ~

1,

that is of Holder

type (Propositions ^3.5, 4.5).

^{Let us} also recall that the

conjecture

which has been left open

by

the above mentioned papers is that the

Doubling Inequality

^{at the Bound-}

ary should hold ^true when the

boundary

^is

Lipschitz. Hopefully,

^{if such}

conjecture

^were proven, then our

stability ^results,

^Theorems

^{2.1, 2.2,} might

be

generalized

^as follows. If I is a

priori

^{known to be}

Lipschitz

^{with a}

sufficiently

^small

Lipschitz

constant, then the

logarithmic stability

^estimates

of Theorems

2.1,

2.2 should

apply

^{also to this case.} If instead the

Lips-

chitz constant of I is

large,

then the best

possible stability

^estimate

might

be worse than

logarithmic.

^{This last}

expectation

is motivated

by

^{the fact}

that two

Lipschitz

surfaces with

large Lipschitz

^constant may be

arbitrarily

close in the sense of the Hausdorff

distance,

^but

locally they

^{need not to}

be

representable

^as

graphs

^{in a common reference}

system (see

^Rondi

[R]

for an

example).

^{If it}

happens

that this is the case for the unknown bound- aries

1,, 12,

^{then it}

might

^also

happen

that estimates on the smallness of

I

ⁱⁿ

QI (in

the Neumann case, for

instance)

^{are worse than}

loga-

rithmic. In fact from the

proofs

^of

Propositions ^{3.2, 4.2,} the importance

^of

proving

^that

^{II, 12}

^are

locally represented

^as

graphs

^{in a common reference}

system

will become evident. This

property

^of

II, 12

will be referred ^to

by saying

^that

^{II, 12}

^{are Relative}

Graphs.

Sufficient conditions

guaranteeing

that the boundaries of the two domains

S22

^{are Relative}

Graphs

^will

be examined in

Proposition

^{3.6. As we}

already mentioned,

ⁱⁿ this paper

we intend to strive after

optimal

results under

possibly

^{minimal a}

priori assumptions

^of

regularity (see i )

^and

iii)

^{in Section}

2). Moreover,

very

general assumptions

^on the unknown

boundary

^{I are made. It} may have ^a

finite,

^but

undetermined,

number of connected components, ^{and no restric-} tion is

placed

^{on their}

topology.

Furthermore we use a

single

measurement

corresponding

^{to one}

boundary ^data, either 1/1

^or g, that can be

prescribed arbitrarily. Concerning

^their

regularity,

^the

assumptions (2.7a), (2.8a)

^are

quite

^{loose and}

essentially correspond

^to the natural ones in the treatment of the direct

problems (1.1), (1.2) respectively.

^In

^addition,

^{we shall re-}

quire

^{a bound on} the oscillation character

(frequency)

^of

1/1

^{or of} g. This is

expressed

^{as a bound on a ratio of two norms: either}

in the Neumann _case,

or

in the Dirichlet case .

Such control will be necessary in order ^to dominate the

vanishing

^{rates of}

the solutions in terms of

quantities

^which

depend only

^{on the}

prescribed

data. Notice that F may be

arbitrarily large,

but it is

expected

^{that the}

constants in the estimates of Theorems

2.1,

^2.2 may deteriorate as F - oo.

The

plan

of the paper is ^as follows.

In Section 2 ^we shall state the main Theorems

2.1, 2.2,

^{we also state}

Corollary

2.3 which

provides

^{a finer}

interpretation

^{of the}

stability

^{estimates in}

the

previous

^theorems.

^Here,

instead of

estimating

the Hausdorff distance of the domains

S21, 522,

^we shall estimate their distance

locally,

^{in terms of the}

graph representation

^{of their}

boundaries,

^{and also}

globally, by viewing

^{them as}

imbedded differentiable manifolds with

boundary.

Sections 3 and 4 contain the

proofs

of Theorem 2.1 and Theorem

2.2, respectively.

^The

proofs

^are

preceded by

^the statements of various

auxiliary propositions (Propositions ^3.1-3.6, Propositions 4.1-4.5).

Section 4 contains also the

proof

^of

Corollary

^2.3.

Section 5 contains the

proof

^{of the}

propositions regarding

the estimates of continuation for

Cauchy problems,

^and

namely Propositions ^{3.1, 3.2,}

4.1 and 4.2.

Such

proofs

^are

accompanied by

^some intermediate lemmas. Lemma 5.1 collects

some

regularity

results for the direct Neumann

problem.

^Lemmas

^5.2,

^{5.3 deal}

with the technical notion of

regularized

^{distance as} introduced

by

^Lieberman.

Section 6 contains the

proofs

^of

Propositions ^3.3,

^4.3

concerning

^estimates

of continuation from the interior.

Section 7 contains all the

proofs concerning doubling inequalities. Namely,

the

proofs

^of

Propositions ^{3.4, 4.4,} dealing

with the interior

doubling inequalities,

the

proofs

^of

Propositions ^{3.5, 4.5,}

^where the results of Adolfsson and Escauriaza

are

adapted

^{to the}

present

purposes. Their result for the Dirichlet

problem

^is

summarized in Lemma 7.1.

Section 8 deals with Relative

Graphs,

^{first in Lemma 8.1 we treat the}

general

^{case of}

Lipschitz boundaries,

^{and we} conclude with the

proof

^of

Propo-

sition 3.6.

2. - The main results

When

representing locally

^a

boundary

^{as a}

graph,

it will be convenient to use the

following

^{notation. For every x E IEBn we shall set x =}

(x’, xn),

^where

x’ E E R.

DEFINITION 2.1. Let S2 be a bounded domain in Given _a, 0

1,

we shall _say that a

portion S

of 8Q is of class

c1,a

with constants po, E &#x3E;

0, if,

for any

P E S,

there exists a

rigid

transformation of coordinates under which

we have P ⁼ 0 and

where (p

^{is a function on C}

RI-1 satisfying

and

REMARK 2.1. To the purpose of

simplifying

^the

expressions

in the various estimates

throughout

the paper, ^we have found it convenient to scale all norms in such a way that

they

^are

dimensionally equivalent

^{to their} argument ^and coincide with the standard definition when the dimensional

parameter

po

equals

^{1. For}

instance,

for any cp ^{E we set}

where

Similarly,

^{we shall set}

and so on for

boundary

^{and trace norms such}

as I i)

^A

priori information

^on the domain.

Our main Theorems

2.1,

2.2 will be based on the

following assumptions

on the domain. Given _{po, M} &#x3E;

0,

^{we assume:}

Here,

and in the

sequel, 10 1

denotes the

Lebesgue

^{measure of S2. We shall}

distinguish

^two

nonempty parts, A,

^{I in a S2 and we assume}

Here,

^{interiors and} boundaries are intended in the relative

topology

ⁱⁿ

Moreover we assume that we can select a

portion

^{within A}

satisfying

^for

some

PI

^{E ~}

and

also, denoting by

^the

portion

^{of of all x E such that}

dist(x, I )

po,

Regarding

^the

regularity

^of

^aS2, given

^a,

^E,

^{0 a}

^1,

^E &#x3E;

0,

^{we assume} that

(2.5)

^{a S2} is of class with constants po, E .

In

addition,

in Theorem

2.1,

^we shall also assume the

following (2.6)

¹ is of class ¹ with constants po, E .

REMARK 2.2. Observe that

(2.5) automatically implies

^{a lower bound on the}

diameter of every connected

component

^{of a S2.}

Moreover, by combining (2.1 )

with

(2.5),

^an upper bound ^on the diameter of S2 can also be obtained. Note also that

(2.1), (2.5) implicitly comprise

^{an a}

priori

upper bound ^on the number of connected

components

^of

i i ) Assumptions

^{about the}

boundary

^data.

Let us set

(that

^is:

APO

^{= We shall assume the}

^following

^on the Neumann data

1/1 appearing

ⁱⁿ

problem ( 1.1 )

and,

^{for a}

given

^{constant F} &#x3E;

0,

Concerning

the Dirichlet

data g appearing

ⁱⁿ

(1.2),

^{we assume}

As noted

already

^{in Remark}

^2.1,

^{norms are}

suitably

^{scaled so to be}

dimensionally equivalent

^{to their}

argument.

i i i ) Assumptions

^{about the}

conductivity.

The

conductivity a

is assumed to be a

given

function from Ilgn with values

n x n

symmetric

^matrices

satisfying

^the

following

conditions for

given

^constants

À,

A, ^{0 h}

1, A &#x3E; 0,

for every

(ellipticity)

for every ^x, y ^E

(Lipschitz continuity)

In the

sequel,

^we shall refer to the set of constants

E,

a,

M, F, À,

^{A as}

to the a

priori

^data.

THEOREM 2.1. Let

521, Q2

^{be two domains}

satisfying (2.1), (2.5).

^Let

Ai, Ii,

i ⁼

1, 2,

^{be the}

corresponding

accessible and inaccessible parts

of their

boundaries.

Let us assume that

A

1 =

A2

⁼

A, 521, S22

^{lie on the same side}

of

^{A and that}

(2.2)- (2.4)

^are

satisfied by

^both

pairs Ai,

^Let

h, 12 satisfy (2.6).

^{Let Ui E be}

the solution to

( 1.1 )

^{when S2 =}

Qi,

^{i =}

1, 2,

^{and let}

(2.7), (2.9)

^be

satisfied. If, given

^E &#x3E;

0,

^{we have}

then we have

where co is an

increasing

continuous

function

^on

[0, oo)

^which

satisfies

and

C,

1], C &#x3E;

0,

⁰ 1] 1 are constants

only depending

^{on the a}

priori

^data.

Here

d1-l

denotes the Hausdorff distance between bounded closed sets of II~n THEOREM 2.2.

Let 521, Q2

^and

Ai, Ii,

^{i =}

l, 2,

^{be as in} Theorem 2.1. Let

(2.1)- (2.5)

^be

satisfied.

^Let Ui ^E be the solution to

(1.2)

^{when S2 =}

Qi,

^{i =}

1, 2,

and let

(2.8), (2.9)

^be

satisfied. If, given

^E &#x3E;

0,

^{we have}

then we have

where (J) is as in

(2.12)

^{and the constants} C, r~, C &#x3E;

0,

⁰ 17 1

only depend

^on

the a

priori

^data.

COROLLARY 2.3. Let the

hypotheses of

either Theorem 2.1 or Theorem 2.2 be

satisfied.

^{There exist} ro, 0 _ro :5 po,

only depending

^on po, E, ^a,

and Eo

&#x3E;

0, only depending

^{on the a}

priori ^data,

^{such that}

if

^E Eo then

for

every P E a

Q

U ^a

Q2

there exists a

rigid transformation of

coordinates under which P ⁼ 0 and

where _(pl, are

c1,a functions

^{on C which}

satisfy, for

every

~8,

0 P

^a,

where

when Theorem 2.1

applies,

when Theorem 2.2

applies,

(J) is as in

(2.12)

^{and K} &#x3E; 0

only depends

^{on E, a and}

(J. Furthermore,

^there

exists a

C 1’a diffeomorphism

^{F : Rn-* such that}

F (Q2)

⁼

S21 and for

every

fi, 0 p

^ot,

with K, ^{lù as above. Here I d :} denotes the

identity mapping.

3. - Proof of Theorem 2.1

Here and in the

sequel

^we shall denote

by

G the connected component ^of

S21 nQ2

^{such that} C

G.

The

proof

of Theorem 2.1 is obtained from the

following

sequence of

propositions.

PROPOSITION 3.1

(Stability

Estimate of Continuation from

Cauchy Data).

^Let

the

hypotheses of

^Theorem 2.1, except

(2.6),

^be

satisfied.

^{We have}

where (J) is an

increasing

continuous

function

^on

^[0, oo)

^which

satisfies

and C &#x3E; 0

depends

^on À, A, E, ^{a and M}

only.

DEFINITION 3.1. Let S2 be a bounded domain in W. We shall _say that

a

portion S

^{of is of}

Lipschitz

class with constants po, E &#x3E;

0, if,

for any

P E S,

there exists a

rigid

transformation of coordinates under which we have P = 0 and

where w

^{is a}

Lipschitz

continuous function on C

satisfying

and

Here the

Co,

¹ norm is scaled

according

^{to the}

principles

stated in Remark

2.1,

that is

PROPOSITION 3.2

(Improved Stability

Estimate of Continuation from

Cauchy Data).

^{Let the}

hypotheses of Proposition

^{3.1 hold}

^and,

ⁱⁿ

addition,

^{let us assume} that there exist L &#x3E; 0 and ro, 0 _{ro po,} such that a G is

of Lipschitz

^{class with}

constants ro, L. Then

(3.1 )

holds with (J)

given by

where _y &#x3E; 0 and C &#x3E; 0

only depend

^on À, A, E, a, M, ^{L and}

po/ro.

We shall denote

PROPOSITION 3.3

(Lipschitz Stability

Estimate of Continuation from the In-

terior).

^{Let S2 be a domain}

satisfying (2.1),

such that 8S2 is

of Lipschitz

^{class with}

constants po, E. Let ^{U E}

H1 (Q)

be the solution to

(1.1), where * satisfies

and, for

^a

given

^{constant F} &#x3E;

0,

and Q

satisfies (2.9).

For every p &#x3E; 0 and _{every Xo} E

Q4p,

^{we have}

where C &#x3E; 0

depends

^{on À,}

A,

E, M, F and

plpo only.

REMARK 3.1. Let us notice that

if 1Jr

^satisfies

(2.7a)-(2.7d),

then it also satisfies

(3.4a)-(3.4c)

up ^to

possibly replacing

^{F with a}

multiple ^cF,

^where

c

only depends

^{on E. In}

^fact,

^for

functions 1/1 satisfying (2.7c)

^the

following equivalence

^{relations can} be obtained

PROPOSITION 3.4

(Interior Doubling Inequality).

^{Let S2 be a domain}

satisfy- ing (2.1 ),

^{such that a Q is}

of Lipschitz

^{class with constants} po, E. Let u E

H (Q)

^be

the solution to

( 1.1 ), where 1/1 satisfies (3.4)

^{and a}

satisfies (2.9).

For every p &#x3E; 0 and _{every Xo} E

Qp,

^{we have}

for

every ^r,

fl

^{s. t. 1}

fJ

^{and 0} p , where C &#x3E; 0 and K &#x3E; 0

depend

^{on À,}

A,

E, M, ^{F and}

p/ po only.

PROPOSITION 3.5

(Doubling Inequality

^{at the}

Boundary).

^{Let Q be a domain}

satisfying (2.1)

^and

(2.5).

^{Let us assume} that the accessible and inaccessible parts

A,

^I

of

^its

boundary satisfy (2.2)-(2.4)

^and

(2.6).

^{Let u E} be the solution

to

(1.1)

^{and let}

(2.7)

^and

(2.9)

^be

satisfied.

^{Let Xo E} I. For any r &#x3E; 0 and _any

{3

&#x3E; 1 we have

where C &#x3E; 0 and K &#x3E; 0

depend

^{on À,}

^{A, E,}

^{M and F}

only.

In the

sequel,

it will be

expedient

^{to introduce a}

quantity

^{which is a}

slight

variation of the Hausdorff distance between

SZ

^{I and}

S22 .

DEFINITION 3.2. We call

modified

distance between

S21

I and

S22

^{the number}

Notice that we

obviously

^have

but,

ⁱⁿ

general, dm

^{does not} dominate the Hausdorff

distance,

and indeed it does not

satisfy

^{the axioms of a distance} function. This is made clear

by

the

following example: ^Ql

⁼

BI (0), S22

⁼

B1 (o) B Bl/2(O).

^{In this case}

dm (S21, S22)

⁼

0,

^whereas

d1t(f2I, S22)

⁼

1 /2.

PROPOSITION 3.6

(Relative Graphs).

^Let

^{521, S22}

^be bounded domains

satisfy- ing (2.5).

There exist numbers

do,

ro,

do

&#x3E;

0,

⁰ ro po,

for

^{which the ratios}

PO

^I

!1l. ^{only depend}

^{on a and E, such that}

^{if we}

^have

-

Po PO

then the

following facts

^hold:

i)

For every P ^E

8 521,

up ^{to a}

rigid transformation of coordinates

^which maps P into the

origin,

^{we have}

where _~p2 are on C

lRn-1 satisfying

for

every

P,

⁰

fl

a , where C &#x3E; 0

only depends

^{on a,}

fl

^{and E.}

ii)

There exists an absolute constant C &#x3E; 0 such that

i i i ) Any

^connected component ^G

of S21 I nQ2

^has

boundary of Lipschitz

^{class with}

constants ro, L, ^where ro is as above and L &#x3E; 0

only depends

^{on a and E.}

PROOF OF THEOREM 2.1. Let us

denote,

^for

simplicity, Let n

&#x3E; 0 be such that

Our first

goal

^{is the}

proof

^{of the}

following inequality

where C &#x3E; 0 and K &#x3E; 0

depend

^on

X, A, E,

^{M and F}

only.

^{As a} pre-

liminary step,

^{let us} show that

(3.15)

^{holds true} when d is

replaced

^with

dm

⁼

Q2),

^the

quantity

introduced in Definition 3.2. Let us assume, with

no loss of

generality,

that there exists _xo E

h

^{C such that}

dist(xo, S22)

⁼

dm.

From

(3.14)

^we

obviously

^have

Suppose ^{now dm}

^po.

By Proposition ^3.5, picking

^{r =}

^dm, P

^{= we have}

where C &#x3E; 0 and K &#x3E; 0

depend

^on

^{X, A, E,}

^{M and F}

only.

^From

(2.5)

^{we can}

find a ball

Br (wo)

^{of radius r =}

compactly

^{contained in}

^SZ

1 f1

2

Hence, applying Proposition

^{3.3 with} p ⁼

r/4,

^{we have}

where C &#x3E; 0

depends

^{on X,}

^{A, E,}

^{M and F}

only.

^From

(3.16)-(3.18)

^we

derive

On the other

hand, when dm &#x3E;

po,

(3.19)

follows from

(3.18)

and from the trivial estimate

with C

only depending

^{on E and M. Hence ~e have}

proved

^that

where C &#x3E; 0 and K &#x3E; 0

depend

^on

^{X, A, E,}

^{M and F}

only.

With no loss of

generality,

^let yo ^E be such that

dist(yo, Q2)

^{= d.}

Let us notice that in

general

yo needs not to

belong

^to

^{a 0 1,}

^{see the}

example

below Definition 3.2. For this reason it is necessary ^to

analyse

various different

cases

separately. Denoting by h

⁼

dist(yo,

^{let us}

distinguish

^the

following

three cases:

where

do is

^the

number

introduced in

Proposition

^3.6.

If case

i )

occurs,

taking

^{zo E 1 such}

that yo - zo - h,

^{we have that}

d -

h ? 2,

^{so that}

^{d 2dm}

^and

^(3.15)

follows from

(3.21).

If case

i i )

occurs, let us set

We have that

By applying Proposition

^{3.4 with r =}

d i , B = d0 ,

i ^{we have}

where C &#x3E; 0 and K &#x3E; 0

depend on X, A, E,

^{a, M and F}

only.

^{Since h} &#x3E;

do

we can

apply Proposition

^{3.3 with p =}

~, ^obtaining

where C &#x3E; 0

depends

^on

^{À, A, E,}

^{a, M and F}

only.

^From

^(3.24)

^and

(3.25)

we have

Let

If 17 fi,

^then

d

i

~ ,

^{so that d =}

^2d

ⁱ

and

(3.15)

follows from

(3.26). If, otherwise,

17 &#x3E;

~,

^then

(3.15)

^follows

trivially,

^{likewise we did in}

(3.20).

If case

i i i )

occurs, then d

do

^and

Proposition

^3.6

applies,

^{so that}

by (3.13)

^and

(3.19)

^we

again

^obtain

(3.15).

Hence, by Proposition ^3.1,

^{we obtain}

where C &#x3E; 0

depends on X, A, E,

^{M and F}

only,

^{whereas K} &#x3E; 0

depends

^on

the same

quantities

^{and in addition on a. Thus we} have obtained a

stability

estimate of

log-log type. Next, by (3.27),

^{we can find} co &#x3E;

0, only depending

on X, A,

E,

a, M and

F,

^{such that} co then d

do. Therefore, by Proposition ^3.6,

^G satisfies the

hypotheses

^of

Proposition

^{3.2. Hence in}

(3.15)

we may

replace

q with where co is as in

Propo-

sition 3.2

(a

modulus of

continuity

^of

log type)

and obtain

(2.11), (2.12).

⁰

4. - Proof of Theorem 2.2 and of

Corollary

^2.3

Here and in the

sequel

^we shall denote

by

G the connected

component

^of

S21 nQ2

^such

that EGG.

The

proof

of Theorem 2.2 is obtained from the

following

sequence of

propositions,

^which

closely parallel Propositions

^3.1-3.5.

PROPOSITION 4.1

(Stability

Estimate of Continuation from

Cauchy Data).

^Let

the

hypotheses of

Theorem 2.2 be

satisfied.

^{We have}

where (J) is an

increasing

continuous

function

^on

[0, oo)

^which

satisfies

where C &#x3E; 0

depends

^{on À,}

^A,

^{E, a and M}

only.

PROPOSITION 4.2

(Improved Stability

Estimate of Continuation from

Cauchy Data).

^{Let the}

hypotheses of Proposition

^{4.1 hold}

^and,

ⁱⁿ

^addition,

^{let us assume}

that there exist L &#x3E; 0 and _ro, 0 _{ro po,} such that a G is

of Lipschitz

^{class with}

constants ro, L. Then

(4.1 )

holds with co

given by

where _y &#x3E; 0 and C &#x3E; 0

only depend

^{onk, A,}

^E,

^{a, M, L and}

po/ro.

PROPOSITION 4.3

(Lipschitz Stability

Estimate of Continuation from the In-

terior).

^{Let S2 be a domain}

satisfying (2.1),

such that a S2 is

of Lipschitz

^{class with}

constants po, E. Let U E

HI(Q)

be the solution to

(1.2),

^where g

satisfies

and, for

^a

given

^{constant F} &#x3E;

0,

and Q

satisfies (2.9).

For every p &#x3E; 0 and _{every Xo} E

Q2p,

^{we have}

where C &#x3E; 0

depends

onk, A, E, M, F and

p/,oo only.

REMARK 4.1. Let us notice that

if g

^satisfies

(2.8a)-(2.8c),

then it also satisfies

(4.4a)-(4.4b)

up ^to

possibly replacing

^{F with a}

multiple ^cF,

^{where c}

only depends

^{on E. In}

^fact,

^for

functions g satisfying (2.8b)

^the

following equivalence

^{relations can} be obtained

PROPOSITION 4.4

(Interior Doubling Inequality).

^{Let SZ be a domain}

satisfy- ing (2.1 ),

^{such that a Q is}

of Lipschitz

^{class with constants} po, E. Let ^{u E} be the solution to

(1.2),

^where g

satisfies (4.4)

^{and a}

satisfies (2.9).

For every p &#x3E; 0 and _{every xo} E

S2 p,

^{we have}

for

every ^r,

f3

^{s.t. 1}

f3

^{and 0}

f3r

p , where C &#x3E; 0 and K &#x3E; 0

depend

^on À, A, E, M, ^{F and}

p/,oo only.

PROPOSITION 4.5

(Doubling Inequality

^{at the}

Boundary).

^{Let S2 be a domain}

satisfying (2.1 )

^and

(2.5).

^{Let us assume} that the accessible and

inaccessible parts ^A,

I

of

^its

boundary satisfy (2.2)-(2.4).

^{Let U E be} the solution to

(1.2)

^and

let

(2.8)

^and

(2.9)

^be

satisfied.

^{Let Xo E I. For} any r &#x3E; 0 and

any P

&#x3E; ^{1 we} have

where C &#x3E; 0 and K &#x3E; 0

depend

^{on À,}

A,

E, a, M and F

only.

PROOF OF THEOREM 2.2.

By using

the trivial estimate

the

proof

is obtained

similarly

^{to the}

proof

of Theorem

2.1,

up ^to obvious

changes.

^D

PROOF OF COROLLARY 2.3. We have that

(2.15)-(2.16)

^follow

immediately

from

(3.12), (2.12)

and either

(2.11) (when

Theorem 2.1

applies)

^or

(2.14) (when

Theorem 2.2

applies).

Next,

^{let us} prove

(2.17).

^{We can find} rl,

h,

⁰ rl ro, 0 h _ro,

only depending

^{on a,}

^E,

^{po, and a number N}

only depending

^{on a,}

^{E, M,}

such that there exist

points Pl

^{E 1 and}

cylinders Cl,

^{I =}

1,..., N,

^centered

at

Pl, having height

2h and basis a

(n - I)-dimensional

disk of radius _ri, such that

UN 1 CI

^{covers both}

^8Qi

^{1 and}

^{a 522,}

^{and each}

^Cl

^{has axis}

along

the direction labeled

by xn

in the local

representation (2.15)

^{when P =}

Pl.

^{Moreover we}

assume

2Cl

C for every I. ^Here

2Cl

denotes the

cylinder

with double sizes and the same center. Notice

that, possibly replacing

^Eo

by

^{a smaller}

number,

^we may ^assume that the

functions Vi

ⁱⁿ

(2.15) satisfy

for every

Let us fix I = 1 and let us define

FI :

^{R" as follows.}

Letting

^{x =}

(x’, xn )

suitable coordinates near P ⁼

Pi,

where

Here: _TJ,

0 ~ ^1,

^{is a smooth} function such that ⁼ 1 when

Ix’l

^rl,

- 0

when 2r,,

^and

i (a , b; ~ ) :

^{R -+ R is an}

uniformly

^smooth

function for every a, b E

[-h/2, h/2] satisfying

for _{every S} E

R,

for _every

and also

t (a, b; b) = a .

Here c 1 is an absolute constant. For instance we can choose r as a suitable

smoothing

^{of the}

piecewise

linear function whose

graph joins (-h, -h), (b, a),

(h, h),

within the square

(-h, h)

^x

(-h, h)

and coincides with the bisector of the first and third

quadrant

^{outside. Now we have}

and hence

One can

verify

^that

FI (Q2nCI)

⁼

Fl (x)

^{= x} for every ^{x E}

aQI naQ2,

F ⁼

FI

^satisfies

(2.17)

and also that if

S22

^is

replaced

^with

FI(Q2),

^then

(2.16)

continues to hold. We _may iterate this

procedure defining inductively analogous

maps

/~

which deform coordinates within the

cylinder ^2Cl

^and

replacing

^{at each}

stage

S22

^with

F’¡(Q2).

^{In the end we set F =}

FN

^{o... o}

Fl.

^{F is an} orientation

preserving ^c1,a diffeomorphism satisfying (2.17)

^{such that =}

a S21

^and also F ⁼ I d outside of the fixed small

neighbourhood

^of

^{a S21} given by U£ 12C’ .

Therefore

F ( S22 )

^{= 0}

5. - Proofs of the estimates of continuation for

Cauchy problems

Throughout

^this

^section,

^let

^{Qi, S22}

^{be two domains}

satisfying (2.1), (2.5).

Let

Ai, Ii, i

⁼

1, 2,

^{be the}

corresponding

accessible and inaccessible parts ^of their boundaries. Let us assume that

A

1 ^-

A2

⁼

A, 521, S22

^{lie on the same}

side of A and that

(2.2)-(2.4)

^{are satisfied}

by

^both

pairs ^{A~ , Ii.}

We shall denote

It is clear that

for every , 4

LEMMA 5.1.

The following

^Schauder type ^{estimates hold}

where C &#x3E; 0

depends

onX, A, E, ^{a and M}

only.